Find if set $I$ of all injective functions $\mathbb{N} \rightarrow \mathbb{N}$ is equinumerous to $\mathbb{R}$

Question

I need to find if set $I$ of all injective functions $\mathbb{N} \rightarrow \mathbb{N}$ is equinumerous to $\mathbb{R}$.
Still, I have no idea how to work with set of functions, it's something new to me since I have only encountered sets of numbers before.

Answer 1

A subset of the injective functions is the set of increasing injective functions. An increasing injective function can be identified with its range, which is just an arbitrary infinite subset of $\Bbb N$. As there are continuum-many subsets of $\Bbb N$ and only countably many of them are finite, we see that there are continuuum-many infnite subsets of $\Bbb N$ and hence at least continuum-many injective functions $\Bbb N\to \Bbb N$.

On the other hand, there are at most continuum-many injective functions $\Bbb N\to\Bbb N$ because their graphs are subsets of $\Bbb N\times\Bbb N$, and as $\Bbb N\times \Bbb N$ is countable and therefore has (only) continuum-many subsets, we are done.